Abstract:
This paper introduces and studies a novel class of equivalence relations in operator theory, called
deformed-metrically equivalent operators. Two operators S and T in B(H) are said to be deformed-metrically equivalent if there exists a positive operator P such that S
∗PS = T
∗
T . This definition generalizes traditional metric
equivalence by incorporating a positive deformation operator P, enabling a richer algebraic and spectral analysis.
We establish several fundamental results, including the preservation of key operator classes such as normality, posinormality, and compactness under suitable commutativity conditions. Spectral inclusion relations are derived under
invertibility assumptions, and the equivalence is shown to be stable under limits, tensor products, and functional calculus. Moreover, the set of all operators deformed-metrically equivalent to a given operator forms an affine space that
is closed in the weak operator topology. These findings deepen the theoretical framework of operator equivalence and
reveal new connections with well-studied classes such as posinormal, supraposinormal, and k-quasi n-power posinormal operators.
T
